Topological convexity in complex surfaces

TL;DR

This paper introduces topologically pseudoconvex (TPC) 3-manifolds in complex surfaces, providing construction tools, classification invariants, and demonstrating that any closed 3-manifold can be TPC embedded in a complex surface with prescribed homotopy class of almost-complex structures.

Contribution

It develops methods for constructing TPC embeddings, classifies almost-complex structures via computable invariants, and shows all closed 3-manifolds can be embedded with any homotopy class in complex surfaces.

Findings

Every closed, oriented 3-manifold admits a TPC embedding in a complex surface.

Invariants classify almost-complex structures on 4-manifolds homotopy equivalent to the 3-manifold.

Two classes of smoothings on product manifolds are realized by holomorphic embeddings.

Abstract

We study a notion of strict pseudoconvexity in the context of topologically (often unsmoothably) embedded 3-manifolds in complex surfaces. Topologically pseudoconvex (TPC) 3-manifolds behave similarly to their smooth analogues, cutting out open domains of holomorphy (Stein surfaces), but they are much more common. We provide tools for constructing TPC embeddings, and show that every closed, oriented 3-manifold M has a TPC embedding in a compact, complex surface (without boundary) realizing any homotopy class of almost-complex structures (the analogue of the homotopy class of the contact plane field in the smooth case). We prove our tool theorems with invariants that classify almost-complex structures on any 4-manifold homotopy equivalent to M. These invariants are amenable to computation and respected by homeomorphisms (not necessarily smooth). We study the two equivalence classes of smoothings on the product of a 3-manifold with a line, and on collared ends. Both classes of smoothings are realized by holomorphic embeddings exhibiting any preassigned homotopy class of almost-complex structures. One class arises from TPC embedded 3-manifolds, while the other likely does not.